Annals of Glaciology 49 2008
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Time dependence of snow microstructure and associated effective thermal conductivity P.K. SATYAWALI,1 A.K. SINGH,2 S.K. DEWALI,1 Praveen KUMAR,1 Vinod KUMAR1 1
Snow and Avalanche Study Establishment (SASE), Manali, Himachal Pradesh 175103, India E-mail:
[email protected] 2 Defence Institute of Advanced Technology, Deemed University, Girinagar, Pune 411025, India ABSTRACT. This paper presents a sequential evaluation of snow microstructure and its associated thermal conductivity under the influence of a temperature gradient. Temperature gradients from 28 to 45 K m–1 were applied to snow samples having a density range 180–320 kg m–3. The experiments were conducted inside a cold room in a specially designed heat-flux apparatus for a period of 4 weeks. A constant heat flux was applied at the base of the heat-flux apparatus to produce a temperature gradient in the snow sample. A steady-state approach was used to estimate the effective thermal conductivity of snow. Horizontal and vertical thick sections were prepared on a weekly basis to obtain snow micrographs. These micrographs were used to obtain snow microstructure using stereological tools. The thermal conductivity was found to increase with increase in grain size, bond size and grain and pore intercept lengths, suggesting a possible correlation of thermal conductivity with snow microstructure. Thermal conductivity increased even though surface area and area fraction of ice were found to decrease. The outcome suggests that changes in snow microstructure have significant control on thermal conductivity even at a constant density.
1. INTRODUCTION In a seasonal snowpack, the possible mechanisms whereby energy can be transported from one region of space to another under the influence of a temperature gradient are: radiation, convection, water-vapour diffusion and thermal conduction (Yosida and others, 1955; Akitaya, 1974). In most practical situations, all these processes to some extent accomplish energy transport, but the relative importance of each contribution can vary markedly. For example, on the surface of a solid material, radiation, conduction and convection are the sole mechanism of heat transport; whereas, inside the body, conduction is the only mechanism possible. Many important materials are of uniform composition throughout, and for them the thermal conductivity is a true physical property of the material, often depending only on the temperature, pressure and composition of the sample. However, in the case of snow, while radiation is the dominant mechanism of heat transfer at the surface, conduction and diffusion also contribute to heat transfer. Therefore, the thermal conductivity is not strictly a property of snow, since it can depend on a large number of parameters, including the history of the material, its development over time and internal structure (Brun and others, 1992; Adams and Sato, 1993; Arons and others, 1994; Arons and Colbeck, 1995; Sato and Adams, 1995; Sturm and others, 1997; Lehning and others, 2002; Schneebeli and Sokratov, 2004; Kaempfer and others, 2005). For this reason, the thermal conductivity is referred to as the effective thermal conductivity (ETC). ETC has been found to vary considerably (Yen, 1962, 1965, 1981; Sturm, 1991; Sturm and Johnson, 1992; Sturm and others, 1997), depending mostly on the snow structure (Schneebeli and Sokratov, 2004). Therefore, the ETC of materials like snow, sand and soil is taken to be the empirical constant of proportionality in the linear relationship between the measured heat transport per unit area and the temperature difference over a prescribed distance in the
material. However, this distinction between homogeneous and inhomogeneous materials is often ignored, leading to confusion, especially where intercomparisons among measurements are concerned (Sturm and others, 1997). In most practical situations, where all three heat-transfer mechanisms are present, the process of measuring the thermal conductivity is greatly complicated. Thus, simple measurements made in the field are inadequate, and it has been difficult to devise measurement methods that unequivocally determine thermal conductivity as a function of controlling processes (Wakeham and Assael, 1999). Many researchers have modelled the thermal conductivity of snow based on its microstructure (Adams and Sato, 1993; Arons and Colbeck, 1998), but these models require fine details about snow before they can be implemented practically. These models were applied to simulate the heat transfer within the snowpack (Brun and others, 1992; Bartelt and Lehning, 2002) with limited success. Recently, Schneebeli and Sokratov (2004) have conducted experiments in a small test device using X-ray computed tomography. The non-destructive experiments were run for 8–12 days under a temperature gradient. Interestingly, they found a slight decrease in ETC in the initial period of metamorphism for fresh snow, followed by an increase. They also measured the structural changes of the snow along with the change in ETC with time. The results varied significantly for low- and highdensity snow. These experiments were untypical, as both ETC and the development of structural changes in snow were measured together. In the present work, experiments were conducted in a specially designed heat-flux apparatus which provided a constant heat flux to the snow samples. ETC of snow was estimated after steady-state temperatures were achieved in the snow samples. A snow block was cut weekly to obtain snow micrographs and determine snow density. Various microstructural parameters were estimated, using the work of Edens (1997), and compared with the evolution of ETC.
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Satyawali and others: Time dependence of snow microstructure
Table 1. Estimation of maximum heat leakage from the heat-flux apparatus for a test sample ETC of test sample
0.09 W m–1 K–1
Measured heat flux at bottom of test sample, A
Measured heat flux at 10 cm height of test sample, B
Average heat losses through side-walls, [(A – B) 100]/A
Average estimated heat losses in sample 4*
Average estimated heat losses in sample 6*
5.64 W m–2
4.18 W m–2
25.9%
20.2 W m–2
16.5 W m–2
*
Using conductivity value of Styrofoam as 0.033 W m–1 K–1 and an average temperature difference of 2.758C between inner and the outer walls of foam.
2. METHODS 2.1. Thermal conductivity
ensuring that most of the heat from the hot plate went into the snow sample.
Fourier’s empirical law simply states that the heat Q transported by conduction per unit area in a particular direction is proportional to the difference of the temperature, T , per unit length in that direction and is given by:
2.2. Design of experimental apparatus
Q ¼ ke T :
ð1Þ
The coefficient of proportionality is known as the ETC and denoted by symbol ke . To maintain a one-dimensional (1-D) heat flow and to minimize radial heat flow in the sample, the snow sample size is taken to be relatively larger than the area over which measurements are made. In practice, it is impossible to arrange an exactly 1-D heat flow in any finite sample, so there will always be a departure from the ideal situation (Wakeham and Assael, 1999). The steady-state method used to measure the ETC of snow often employs the geometry of parallel plates. The heat-flux apparatus designed for the present work is shown in Figure 1a. The snow sample of height d is placed over the base (copper) plate. A small amount of heat (Q) is generated electrically at the base plate having an area A, and the heat (Q) is transported through the sample to the snow surface. The temperatures of the snow surface and hot (copper) plate are measured very precisely, as is the electric input of energy, so that the ETC can, in principle, be evaluated from: T Q ¼ Ake : ð2Þ d The electric energy generated at the hot plate is assumed to be conducted through the snow surface. Although the insulation around the snow sample was enough, it was difficult to prevent spurious heat losses. However, to ensure 1-D heat flow, a larger snow sample size was taken in our experiments. Temperature was measured at different depths of the snow sample in the horizontal plane (see section 2.2),
A heat-flux apparatus having d.c. heaters at the base was designed with a view to conducting the temperature gradient experiments under controlled conditions. For this purpose, a high-resistance heater wire was evenly distributed over the entire base area to prevent any temperature variation. The heater was mounted on top of wooden blocks with a Bakelite base and insulation so that the bottom was insulated and all heat was trapped inside. The total wattage of the base heater was kept at 50 W for an area of 1.0 0.3 m2 powered with a 12–24 V d.c. source. A plate of electrolytic grade copper with a base area of 1.0 0.3 m2 and thickness of 6 mm (to sustain the weight of the snow sample) was kept above the heater, with a small air gap between the heater and copper plate, which helped in the heat transfer to the copper plate by radiation and convection. This achieved a slow and uniform rise in temperature and prevented the formation of hot spots on the copper plate. Six RTD-type temperature sensors (PT-100) were flushed within the copper plate to monitor its temperature. A schematic of the experimental set-up is shown in Figure 1b. Before the experiments started, a uniform temperature distribution over the entire copper plate was ensured. The temperature of the copper plate was found to be within 0.558C of the mean. Sufficient insulation (100 mm Styrofoam) was provided at the bottom and sideways to reduce heat losses, and 1-D heat flow was maintained. In our case, 3% of the total heat generated by the heater escaped downwards from the bottom of the apparatus. This was determined by finding the difference between the amount of heat produced by the heater and the heat input to snow. To measure the heat losses from the side-walls, two heat-flux sensors were placed at the base plate, and two exactly above
Table 2. Experimental details of all the tests conducted on various snow samples in cold room Temperature gradient experiment Total days of experiment Temperature of hot (copper) plate (8C) Temperature of snow surface (cold) (8C) Average temperature gradient (8C cm–1) Average snow temperature (8C) Heat flux at base (W m–2) Snow sample height (cm) Snow density before experiment (kg m–3) Snow density after experiment (kg m–3) % increase in snow density Chamber temperature (8C)
Sample 3
Sample 4
Sample 5
Sample 6
28 –5.4 –14.0 0.46 –9.7 5.5 18.5 250 265 6 –14.0
28 –4.4 –10.0 0.28 –7.2 4.5 20 320 320 0 –10.0
28 –8.1 –17.0 0.45 –12.6 5.5 20 310 315 2 –17.0
28 –4.5 –10.0 0.28 –7.3 5.5 20 180 200 11 –10.0
Satyawali and others: Time dependence of snow microstructure
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these sensors at a height of 10 cm from the base plate. The average heat loss through the side-walls was found to be 25.9%. Schneebeli and Sokratov (2004) also observed heat losses through vertical walls as high as 22%, for a smaller sample. Table 1 shows the heat losses from the apparatus as obtained for a test sample. The temperature distribution was recorded laterally at a height of 10 cm along the length of the snow sample. A temperature gradient of 1.58C m–1 was found, which indicates that there was probably some heat flow in the horizontal direction. A robust low-temperatureapplication data acquisition system (Model DT-800, DataTaker, Australia) was used, which was interfaced with a heat-flux unit for logging of temperatures and heat fluxes.
2.3. Experiments Sieved natural snow was used for all the experiments. Snow was collected from Patsio (3800 m a.s.l.) research station, transported by air to the cold laboratory in insulated boxes and stored at –208C. This snow was composed of fine grains with an average diameter up to 150 mm. The heat-flux apparatus was filled with the snow to a height of 0.20 m by using a 1.0 mm sieve opening. Four sets of experiments were conducted in the heat-flux apparatus to monitor the structural changes in the snow sample due to the effects of temperature and temperature gradient. The thermal conductivity of snow was estimated using steady-state methods (Equation (2)). These experiments were conducted for 4 weeks each, and snow micrographs of vertical and horizontal surface sections were obtained on a weekly basis. The base heater was switched off 1 day before the snow sample was cut for the surface section preparation and density measurement. A snow sample of 0.2 m along the length of the snow sample was cut for the analysis. In this way, a sample size of 0.30 0.20 0.20 m3 was available each time for the thermal conductivity and microstructure analysis. The gap created by cutting the snow sample was refilled with similar snow by sieving. Details of all the experiments are given in Table 2.
2.4. Microtoming In a cold room at –108C, a smaller snow sample (3 3 3 cm3) was cut from the bottom portion of the snow sample for microtoming. Dimethyl phthalate, a liquid pore filler, was poured around the sample until the liquid completely filled the pore space. The dimethyl phthalate was allowed to freeze overnight before sectioning the snow sample. Surfacesection samples were made following the procedure of Perla (1982). Microtoming was done in an automated polycut machine (Leica, Germany). Micrographs of horizontal and vertical surface sections were taken from the bottom half of the snow sample. In this way, a total of ten micrographs were taken for each sample. The micrographs were taken along with the measurement scale for image calibration so that details of each sample could be compared. These images were essentially greyscale and could easily be made into a binary for image processing.
2.5. Image processing The digital image in Tagged Image File Format (TIFF) was loaded into the Image-Pro Plus (IPP) 4.0 software. Image information was adjusted using a greyscale threshold and then converted into a binary format. This is a crucial task for any user, as threshold limits cannot be defined precisely. Thresholding will always add some undesired portion or
Fig. 1. (a) Photograph of heat-flux apparatus filled with snow. (b) Schematic of the experimental set-up (top and front views).
remove some important features, especially the connecting points on a micrograph. Utmost care was taken to manually adjust the connection between the grains. Since IPP software can edit the images, almost all micrographs were examined and modified before applying stereological analysis methods (Underwood, 1970). The image was finally saved in PICT format as input to a custom stereological analysis program (Edens, 1997). With the help of this program we determined the sample microstructural parameters. Once the image is sufficiently processed, Edens’ program (Edens, 1997) can calculate sample geometrical statistics from the digital images. This program calculates the mean intercept length (ice and pore), three-dimensional (3-D) grain and bond radius, area fraction of ice and specific surface area from a PICT image file. The software program assumes that the solid inclusion is spherical. The definition of grain bond given by Kry (1975) has been used in the current work. According to this, accepted definition for the bond to exist, a minimum constriction ratio of 0.7 must exist between connected segments of ice in a surface section plane. This means the maximum bond radius can reach up to 30% of the grain radius. Later, Edens (1997) found that a higher constriction ratio (i.e. bond radius
